Rates of memory loss for null recurrent Markov chains

Abstract

Orey (1962) proved that for an irreducible, aperiodic, and recurrent Markov chain with transition operator $P$, the sequence $P^n(\hat{I}¼-\hat{I}½)$ converges to zero in total variation for any two probability measures $\hat{I}¼$ and $\hat{I}½$. In other words, all such Markov chains exhibit memory loss. While the rates of memory loss have been extensively studied for positive recurrent chains, there is a surprising lack of results for null recurrent chains. In this work, we prove the first estimates of memory loss rates in the null recurrent case.